Math Problem Statement

\sqrt((30)/(\sqrt(10)))

Solution

Let's simplify the expression:

3010\sqrt{\frac{30}{\sqrt{10}}}

Step 1: Simplify the denominator. 10 is in the denominator, so we will rationalize it by multiplying both the numerator and denominator by 10.\sqrt{10} \text{ is in the denominator, so we will rationalize it by multiplying both the numerator and denominator by } \sqrt{10}.

3010×1010=301010\frac{30}{\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}} = \frac{30 \sqrt{10}}{10}

Step 2: Simplify the fraction. 301010=310\frac{30 \sqrt{10}}{10} = 3 \sqrt{10}

Now the expression becomes: 310\sqrt{3 \sqrt{10}}

Step 3: Simplify inside the square root. We can break it down as: 310=310=3101/4\sqrt{3 \sqrt{10}} = \sqrt{3} \cdot \sqrt{\sqrt{10}} = \sqrt{3} \cdot 10^{1/4}

This is the simplified form: 3101/4\sqrt{3} \cdot 10^{1/4}

Would you like more detailed steps or explanations on any part of this?

Here are some questions you might find interesting:

  1. How do you rationalize square roots in fractions?
  2. Can this method be applied to cube roots or higher powers?
  3. What other methods are there to simplify radical expressions?
  4. How do you simplify nested square roots like the one in this problem?
  5. Why is rationalizing the denominator important in certain mathematical contexts?

Tip: When simplifying expressions with square roots, it's often useful to rationalize the denominator to make the expression easier to work with.

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Math Problem Analysis

Mathematical Concepts

Radical Expressions
Rationalizing Denominators
Square Roots

Formulas

\sqrt{a} \times \sqrt{b} = \sqrt{a \cdot b}
Rationalizing the denominator: \frac{a}{\sqrt{b}} \times \frac{\sqrt{b}}{\sqrt{b}} = \frac{a \sqrt{b}}{b}

Theorems

Properties of Square Roots
Rationalizing the Denominator

Suitable Grade Level

Grades 9-11